anonymous
  • anonymous
do these lines intersect? [x,y,z]=[2,-3,2]+s[3,4,-10] [x,y,z]=[3,4,-2]+s[-4,5,3]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Do you want me to show you what I did?
anonymous
  • anonymous
because I'm not getting the answer my book has
anonymous
  • anonymous
alright. lets see what you did

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anonymous
  • anonymous
So I made the parametric equations for both lines line 1 : x=2+3s y=-3+4s z=2-10s line 2: x=3-4t y=4+5t z=-2+3t
anonymous
  • anonymous
Then I set the equations equal to each other. The x of line 1 with the x of like two and so on. I got these 3 equations 3s+4t=1 4s-5t=7 -10s-3t=-4
anonymous
  • anonymous
yep
anonymous
  • anonymous
then I used elimination on 3s+4t=1 and 4s-5t=7. I multiplied 3s+4t=1 by 4 and 4s-5t=7 by 3. I got t=-17/31 after solving
anonymous
  • anonymous
I then subbed t=-17/31 into 4s-5t=7 to find s. I got s=33/31
anonymous
  • anonymous
yep
anonymous
  • anonymous
After getting t and s. I subbed them into -10s-3t=-4 to see if the right side is equal to the left side. I got -9=-4
anonymous
  • anonymous
but my book has an answer but since I'm given x, y and z. the the point of intersection should also have an x y and z in it. but it gives (33/31, -17/31). But those are my s and t values. so I'm a little confused
anonymous
  • anonymous
because the question asks if the intersect, and what the points are if they do
anonymous
  • anonymous
do you think they intersect?
anonymous
  • anonymous
no, because the left side doesn't equal the right side. Also they are not parallel. So I think they are skew lines. But i think i'm wrong
anonymous
  • anonymous
alright. i would have done it a little different to you but here is what i did.
anonymous
  • anonymous
don't both about your s and t values as i chose s and t values for the opposite lines
anonymous
  • anonymous
so yea i got [3t-4s,4t-5s,-10t-3s]=[1,7,-4]
anonymous
  • anonymous
ok. now here is the thing
anonymous
  • anonymous
i'll say it later, let me get it into parametric form.
anonymous
  • anonymous
so yea your right with parametric form
anonymous
  • anonymous
3t+4s=1 4t-5s=7 -10t-3s=-4
anonymous
  • anonymous
for simplicity sake let X represent 3t+4s=1 Y represent 4t-5s=7 and Z yea u get the drift.
anonymous
  • anonymous
Now for these two lines to intersect, then when solving simultaneously with X and Y these solution should also be the same for when you solve for X and Z or Y and Z You only need to do either 2 solvings. like say X and Y and X and Z or X and Y and Y and Z
anonymous
  • anonymous
So you have solutions for X and Y which were t=33/31 and s=-17/31
anonymous
  • anonymous
now for X and Z we get solutions t=13/31 and s=-2/31
anonymous
  • anonymous
since we get totally different solutions for t and s in both scenario's then these lines do not intersect.
anonymous
  • anonymous
to check this we can plug these numbers into [3t-4s,4t-5s,-10t-3s]=[1,7,-4] and you will see that you won't get [1,7,-4]
anonymous
  • anonymous
got any queries?
anonymous
  • anonymous
so why does my book give me an answer that is the s and t values. Because before in other question it would just say no and that was all
anonymous
  • anonymous
what do you mean by the s and t values?
anonymous
  • anonymous
the answer my book gives is (33/31, -17/31)
anonymous
  • anonymous
the s and t values i get
anonymous
  • anonymous
thats weird. those values only seem to work of X and Y
anonymous
  • anonymous
either we are wrong or the book is wrong.
Zarkon
  • Zarkon
1) double check that you have the question written correctly...2) if you do then the book is wrong
anonymous
  • anonymous
my book always numbers the equations and they always use the first two, x and y then sub into z to see if the sides are equal
anonymous
  • anonymous
i was taught differently. but same concept still applies.
anonymous
  • anonymous
yea so see if you wrote the question down correctly.
anonymous
  • anonymous
yea, i wrote is right
anonymous
  • anonymous
kk. book is silly,
anonymous
  • anonymous
I want to ask my teacher, but i'm scared to say that the book is wrong and i'm right, since i'm just learning this and i'm not the best person in math
anonymous
  • anonymous
doesn't hurt to go through your method with your teacher.
anonymous
  • anonymous
ok. could to stay around. I might have another question
anonymous
  • anonymous
k
anonymous
  • anonymous
I have another question .
anonymous
  • anonymous
My book asks if line r=r(with arrow on top) [-5,1,-2]+k[1,6,5] intersects with the equation [x,y,z]=[2,3,-1]+s[1,3,4]+[-5,4,7]. I know that instead of [x,y,z] i can put r (with arrow on top) but what is with the notation of the first equation, does it mean anything else
anonymous
  • anonymous
can you draw it for me- just the r arrow thing
anonymous
  • anonymous
|dw:1371014894598:dw|
anonymous
  • anonymous
sorry for my bad drawing
anonymous
  • anonymous
thats what it has
anonymous
  • anonymous
thats just a notation for a vector line
anonymous
  • anonymous
so don't get overwhelmed by it. doesn't contribute to your calculations
anonymous
  • anonymous
but why does it have r=r with arrow
anonymous
  • anonymous
i just put it r with arrow = blah blah
anonymous
  • anonymous
because r itself could be a point but since its r dash then this represents vector notation.
anonymous
  • anonymous
ok thank you so much for your help
Zarkon
  • Zarkon
You can write vectors on the site \[\vec{r}\] \vec{r}

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